In 1949, Onsager [1]} conjectured that
every weak solution of the incompressible homogeneous Euler equations with Hölder continuity exponent \(\alpha >1/3\)  must conserve energy and there exists a weak solution with energy dissipation to the Euler equations when \(\alpha <1/3\) . Let \(v\) 
represent the fluid velocity field and \(\pi \)  stand for the scalar pressure.
The classical incompressible Euler equations describing inviscid fluids read
\(\left\lbrace \begin{aligned}& v _{t} +\mathrm {div}\,( v\otimes v)+\nabla \pi =0,\\&\mathrm {div}\,v=0,\\& v(x,0)=v_{0}.\end{aligned}\right.\) 
